Invalid argument forms

 

 Consider the following argument form:

p

q

Therefore r

If we let p be 'It is raining in the southeast', let q be 'increased rain usually helps crops produce a higher crop yield' and r be 'crops in California will produce more' then the resulting argument is not valid (check to make sure you see a possible way to have all true premises and a false conclusion).

On the other hand, if we let p be 'If travelers always arrive at their destinations excited but tired then the central time zone is one hour behind the eastern time zone', q be 'travelers always arrive at their destinations excited but tired' and r be 'the central time zone is one hour behind the eastern time zone' then we have the exact same argument as given in Example 3.0.1, and thus is a valid argument. This example tells us that some argument forms can result in arguments which are not valid or arguments which are valid depending on the propositions used to replace the letters used in the argument form. This outcome is impossible for valid argument forms. Valid argument forms always produce valid arguments irrespective of the propositions chosen to replace the variable letters used in the argument form.

This leads to the following definition.

Definition: Let p, q, r, etc. stand for propositions. An invalid argument form is an argument given in terms of p, q, r, such that the resulting argument may be invalid or may be valid depending on the propositions used to replace the variables p, q, r, etc.

 

Notice that the definition for an invalid argument form is just the negation of the definition of "valid argument form". The surprise occurs when we negate the phrase, "… the resulting argument is always valid for any choice of propositions for p, q, r etc." where the negation of "always" is "at least one is not", which together with the above observation leads to the given definition.

There are many invalid argument forms. However some invalid forms are very similar to valid forms and such similarity historically mislead some to think the resulting form was actually valid. These forms have been traditionally called formal deductive fallacies, but for this text we will use the more descriptive term 'pseudo-valid argument form' to emphasize their similarity to valid argument forms.

Definition: A pseudo-valid argument form is an invalid argument form which is similar in form to valid argument forms.

Here we give a small list of pseudo-valid argument forms, comparing them to valid argument forms given on the left.

 

Valid and pseudo-valid argument forms

Valid argument form

Pseudo-valid argument form

modus ponens/affirming the antecedent

If p then q

p

Therefore q

 

denying the antecedent

If p then q

not p

Therefore not q

 

modus tollens/denying the consequent

If p then q

not q

Therefore not p

 

affirming the consequent

If p then q

q

Therefore p

 

disjunctive syllogism / process of elimination

p or q

not p

Therefore q

 

false dilemma

p or q

p

Therefore not q

 

 

 

Due to the similarity between the valid form and the pseudo-valid form, historically some people assumed the pseudo-valid from was valid. This error in reasoning many times is called a deductive fallacy.

There is one technicality we should note here that applies to both disjunctive syllogism and false dilemma. The word 'or' has two different meanings in English. Those meanings are given the names 'inclusive or' and 'exclusive or'.

Two different meanings of 'or'

 

The phrase 'inclusive or' means 'p and/or q'. Specifically given two propositions p and q, then the resulting proposition 'p or q' means 'p or q' is a true proposition even when both p and q are true. An example would be, "John is a student or John is a father'. If John is both a student and a father, then the entire sentence is still true.

 

The phrase 'exclusive or' means 'p or q but not both'. Specifically given two propositions p and q, then the resulting proposition 'p or q' means 'p or q' is a true proposition only when exactly one of p or q is true. An example would be, "Today is Monday or today is Wednesday'. Here the meanings of the weekdays exclude the possibility that both p and q can be true and the sentence is true only when it is really Monday or really Wednesday.

 

The word 'or' will always be understood to be the inclusive version. When we want to refer to the exclusive or, we will write 'xor' or use the additional phrases like, 'p or q but not both', or 'exactly one of p or q is true'.

 Let's examine the concept of argument form in more detail by comparing two different arguments, one has a valid form while the other has an invalid form.

Challenge: Verify that the following argument form is valid

(which means it is impossible to have all true premises and a false conclusion).

 

p xor q

p

Therefore not q

 

 

 

Example 3.1.1

(a)

If it rains, then the game will be cancelled.

It is raining.

Therefore the game will be cancelled.

 

(b)

If it rains, then the game will be cancelled.

It did not rain.

Therefore the game will not be cancelled.

 

Notice that (a) is an example of modus ponens, and hence must be valid.

However example (b) is an instance of denying the antecedent, so the resulting argument form may be valid or invalid. How do we know which? For now we can use the two step method for invalidity.

One final point before you get a chance to test your understanding. Since pseudo-valid argument forms are forms which were thought to be valid forms, when an argument has such a form and is indeed invalid, we will classify it as an erroneous proof in our classification scheme for arguments.

Test your understanding: determine the argument form

 

Determine the argument form of the following arguments.